Question

the value of( 52 + 6√43) whole power 3 by 2 - (52 - 6 root 43) whole power 3 by 2 please answer

Answer 1

Please see the attachment

Answer 2

Answer:

The value of  $(52+ (2)\sqrt{43(9)} )^{\frac{3}{2} } - (52-(2)\sqrt{43(9)} )^{\frac{3}{2} }$ is  828.

Step-by-step explanation:

Given that $(52 + 6\sqrt{43} )^{\frac{3}{2} } - (52-6\sqrt{43})^{\frac{3}{2} }$

($(52+ (2)(3)\sqrt{43} )^{\frac{3}{2} } - (52-(2)(3)\sqrt{43} )^{\frac{3}{2} }$

$(52+ (2)\sqrt{43(9)} )^{\frac{3}{2} } - (52-(2)\sqrt{43(9)} )^{\frac{3}{2} }$

We know that (a+b)³ = a³ +b³+3a²b+3ab²

(a-b)³ = a³ -b³ -3a²b+3ab²

(a+b)² = a²+2ab+b²

(aⁿ)ˣ = aⁿˣ

$\sqrt{(52+6\sqrt{43})^{3} }$ = $\sqrt{((43+9)+2(3)\sqrt{43})^{3} }$

                       = $\sqrt{((\sqrt{43})^{2}+(3)^{2}+2(\sqrt{43})(3))^{3} }$

                      =$\sqrt{(((\sqrt{43)}+3)^{2})^{3} }$

                     =$[(\sqrt{43} +3)^{2} ]^{\frac{3}{2} }$

                     =(√43+3)³           ---------------------------(1)

$\sqrt{(52-6\sqrt{43})^{3} }$ = $\sqrt{((43+9)-2(3)\sqrt{43})^{3} }$

                       = $\sqrt{((\sqrt{43})^{2}+(3)^{2}-2(\sqrt{43})(3))^{3} }$

                      =$\sqrt{(((\sqrt{43)}-3)^{2})^{3} }$

                     =$[(\sqrt{43} -3)^{2} ]^{\frac{3}{2} }$

                     =(√43-3)³     ---------------------------(2)

From the question equation(2) subtracting from equation(1)

That is

$(52+ (2)\sqrt{43(9)} )^{\frac{3}{2} } - (52-(2)\sqrt{43(9)} )^{\frac{3}{2} }$=   (√43+3)³ - (√43-3)³

=[(√43)³+(3)³+3(√43)²(3)+3(√43)(3)²] - [(√43)³ -(3)³-3(√43)²(3)+3(√43)(3)²]

=(√43)³+(3)³+3(√43)²(3)+3(√43)(3)² - (√43)³ +(3)³+3(√43)²(3)-3(√43)(3)²]

=3×43×3+27+3×43×7+27

=387×2+54

=774+54

= 828.

Therefore, $(52+ (2)\sqrt{43(9)} )^{\frac{3}{2} } - (52-(2)\sqrt{43(9)} )^{\frac{3}{2} }$= 828.